Correlations for Nusselt Number in Free Convection...

American Journal of Mechanics and Applications 2015; 3(2): 8-18

Published online May 4, 2015 (http://www.sciencepublishinggroup.com/j/ajma)

doi: 10.11648/j.ajma.20150302.11

ISSN: 2376-6115 (Print); ISSN: 2376-6131 (Online)

Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation

Jasim Abdulateef1, Ayad Hassan

2

1Mechanical Engineering Department, Diyala University, Diyala, Iraq 2Material Engineering Department, University of Technology, Baghdad, Iraq

Email address: [email protected] (J. Abdulateef), [email protected] (A. Hassan)

To cite this article: Jasim Abdulateef, Ayad Hassan. Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical

Simulation. American Journal of Mechanics and Applications. Vol. 3, No. 2, 2015, pp. 8-18. doi: 10.11648/j.ajma.20150302.11

Abstract: This paper presents a numerical study on three-dimensional transient natural convection from an inclined

isothermal square plate. The finite difference approach is used to solve the governing equations, in which buoyancy is modeled

via the Boussinesq approximation. The complete Navier-Stokes equations are transformed and expressed in term of vorticity

and vector potential. The transformed equations are solved using alternating direction implicit (ADI) method for parabolic

portion of the problem and successive over relaxation (SOR) for the elliptic portion. Solutions for laminar case are obtained up

to Grashof number of 5x104 as well as the inclination angles were varied from 0

o to 180

o with 30

o intervals, and the Prandtl

number of 0.7 is considered. The results are shown in terms of isothermal plots, and the local and average Nusselt numbers are

also presented. The simulation results show that the main process of heat transfer is conduction for Grashof number less than

103 and convection for Grashof number larger than 10

3. It is also found that, the values of Nusselt number show fairly large

dependence on inclination angle and there is a significant difference in heat transfer rates between the upward and downward

orientation. The average Nusselt number increases to 20% at the vertical position compared to horizontal position then

decreases with increasing inclination of plate at downward orientation. Based on the results obtained, correlations have been

proposed to evaluate the Nusselt numbers of both upward and downward orientation. Validations of the present results are

made through comparison with available numerical and experimental data, and a good agreement was obtained.

Keywords: Natural Convection, Laminar Flow, Vorticity and Vector Potential, Inclined Isothermal Plate

1. Introduction

Recently, the clarification of flow and heat transfer

mechanisms of natural convection in square plates-which is

often observed in many science and engineering applications

such as in printed circuit board (PCB), solar cell, and thin film

manufacturing chamber - is very important for both industrial

applications and academic research [1].

The majority of the studies available in the literature on

natural convection heat transfer from horizontal rectangular

plates facing upwards have been carried out experimentally,

by using air or water as working fluid [2-10]. Al-Arabi and

El-Riedy [2] obtained local and average heat transfer data for

horizontal plates of different shapes on the basis of the amount

of steam condensate collected from compartments underneath

the plates. Goldstein and Lau [3] performed mass transfer

experiments with square naphthalene plates in air. Further

heat transfer results for rectangular surfaces were then found

by Lewandowski and Kubski [4]. Kitamura and Kimura [5]

conducted an experimental study of natural convection in

water from slender rectangular plates heated with uniform

heat flux and equipped with fences at both longer sides to

inhibit side flows, thus obtaining a two-dimensional flow field

over the plates. Fluid flows and heated surface temperatures

were visualized by dye tracers and liquid-crystal thermometry,

respectively. The results showed that, four distinct flow

regions appeared, consisting of a two-dimensional laminar

boundary layer region near the leading edges, a transitional

region characterized by a three-dimensional flow separation

and by the attachment of ambient fluid onto the heated surface

downstream of the flow separation, a fully turbulent region,

and a collision region near the plate centerline. Lewandowski

9 Jasim Abdulateef and Ayad Hassan: Correlations for Nusselt Number in Free Convection from an Isothermal Inclined

Square Plate by a Numerical Simulation

et al. [6] performed heat transfer measurements and visualized

the convective flow structures in water above rectangular

plates by injecting a dark-blue dye tracer through small holes

distributed along the perimeter of each plate. In the same year,

Pretot et al. [7] performed experiments using a rectangular

plate heated with uniform heat flux. It’s found that, starting

from the edges of the plate, the heat transfer is characterized

by a thermal boundary layer that breaks away from the surface

near the center of the heated surface, giving rise to a buoyant

thermal plume. Rafah Aziz [8] studied experimentally natural

convection heat transfer from inclined isothermal square flat

surface. Heat transfer measurements from rectangular plates

facing upward were carried out in air by Martorell et al. [9],

and in water by Kozanoglu and Lopez [10].

Several theoretical studies have been carried out to evaluate

natural convection heat transfer from isothermal plates. The

first numerical solutions of the full Navier-Stokes, continuity

and energy transfer equations for natural convection from

horizontal upward-facing plates were carried out by Goldstein

and Lau [3]. Chen et al. [11] considered the upward facing

horizontal semi-infinite plate with uniform wall temperature

or uniform heat flux as a special case of a larger

boundary-layer investigation of natural convection from

inclined plates with variable surface temperature or heat flux.

Lewandowski [12] presented a quasi-analytical solution of a

theoretical model, in which the boundary layers grow from

each plate-edge up to transforming into a plume at the

separation point. Wei et al. [13] presented a numerical

solutions of the mass, momentum and energy governing

equations on two-dimensional natural convection from

isothermal plates heated at both sides. Martorell et al. [9],

derived two and three dimensional solutions for slender plates,

concluding that although the flow near the ends of the plate is

typically three-dimensional, the overall flow suffers no

dramatic change with respect to the two-dimensional structure.

Hassan [14] carried out a numerical study on three-

dimensional natural convection from inclined discs and rings

at constant temperature. He also presented a numerical

solution of natural convection from horizontal plate at

constant wall temperature [15]. A review of the heat transfer

data for free convection from upward facing heated

rectangular plates is conducted by Corcione [16].

However, information and data for three-dimensional,

transient natural convection from inclined square surface

appears to be limited. The objective of this study is to develop

a numerical simulation model to investigate the

three-dimensional transient laminar natural convection from

isothermal square plate subject to the influence of orientation.

The Grashof numbers and inclination angles of the square

plate are ranged from 103 to 5x10

4 and from 0

o to 180

o,

respectively. As far as the fluid characteristics, the Prandtl

number of 0.7 was considered. The finite difference approach

is used to solve the governing equations.

2. Governing Equations of the Present

Study

The basic equations describing the flow driven by natural

convection consist of mass conservation, momentum and

energy. These equations are solved numerically by finite

difference approach. The numerical simulation of the problem

was performed using a computer program written by

FORTRAN. The Boussinesq approximation was included in

the buoyancy force term so that all densities are assumed to be

constant in the body force except the one in gravity term

which changes with temperature.

The appropriate equations for an incompressible Newtonian

fluid are [17]:

0=⋅∇ Vv

(1)

VPgDt

VD vvv

2∇+∇−= µρρ (2)

Θ∇=Θ⋅ 2K

Dt

Dcp ρ (3)

The density is given by linearized state equation as

followed [17]:

))(1( ∞∞ −−= TTβρρ (4)

Equation (4) is used in buoyancy term of the momentum

equation, whereas ρ=ρ∞ is used elsewhere where ρ∞ is the

density at reference condition. Using Boussinesq

approximation, the governing equations for constant fluid

property can be expressed as:

0=⋅∇ Vv

(5)

VPgDt

VD vvv

2)( ∇+∇−Θ−Θ−= ∞∞ µβρρ (6)

Θ∇=Θ⋅ ∞2

KDt

Dcp ρ (7)

where (p) represents the perturbation of static pressure from

the hydrostatic value.

3. Mathematical Model

3.1. Problem definition

The three-dimensional geometry being solved in this study

is shown in Figure1. The energy from the isothermal plate

facing upward or downward orientation is dissipated by

convection of ambient fluid at T∞. The length of square plate is

H and the surface is inclined with an angle of Φ with the

horizontal axis.

American Journal of Mechanics and Applications 2015; 3(2): 8-18 10

Figure 1. Geometry of the case study

3.2. Mathematical Model

The continuity, momentum and energy equations for three-

dimensional laminar flows of an incompressible Newtonian

fluid are written. Following assumptions are made: there is no

viscous dissipation, the gravity acts in the vertical direction,

fluid properties are constant and fluid density variations are

neglected except in the buoyancy term (the Boussinesq

approximation) and radiation heat exchange is negligible. The

governing equations are obtained as follows:

Continuity equation

0=∂∂+

∂∂+

∂∂

z

w

y

v

x

u (8)

Three momentum equations

x-component

)(

1)sin()(

2

2

2

2

2

2

z

u

y

u

x

u

dx

dpg

z

uw

y

uv

x

uu

t

u

∂∂+

∂∂+

∂∂+

−ΦΘ−Θ=∂∂+

∂∂+

∂∂+

∂∂

∞∞

υ

ρβ

(9)

y-component

)(1

2

2

2

2

2

2

z

v

y

v

x

v

dy

dp

z

vw

y

vv

x

vu

t

v

∂∂+

∂∂+

∂∂+−=

∂∂+

∂∂+

∂∂+

∂∂

∞

υρ (10)

z-component

)(

1)cos()(

2

2

2

2

2

2

z

w

y

w

x

w

dz

dpg

z

ww

y

wv

x

wu

t

w

∂∂+

∂∂+

∂∂+

−ΦΘ−Θ=∂∂+

∂∂+

∂∂+

∂∂

∞∞

υ

ρβ

(11)

Energy equation

)(2

2

2

2

2

2

zyxzw

yv

xu

t ∂Θ∂+

∂Θ∂+

∂Θ∂=

∂Θ∂+

∂Θ∂+

∂Θ∂+

∂Θ∂ α (12)

3.3. Nondimensionalization

Before the discretization process, we introduce the

following dimensionless variables:

H

xX = ,

H

yY = ,

H

zZ = ,

αuH

U = ,αvH

V = ,α

wHW = ,

2H

tατ = , 2)/( H

pP

αρ ∞

= ,∞

∞

Θ−ΘΘ−Θ

=w

T

In terms of these dimensionless variables, the

non-dimensional governing equations are obtained as:

Continuity equation

0=∂∂+

∂∂+

∂∂

Z

W

Y

V

X

U (13)

Three momentum equations

X-component

)(

)sin(

2

2

2

2

2

2

Z

U

Y

U

X

U

dX

dPRa

Z

UW

Y

UV

X

UU

U

∂∂+

∂∂+

∂∂+

−Φ=∂∂+

∂∂+

∂∂+

∂∂

υ

τ (14)

Y-component

)(2

2

2

2

2

2

Z

V

Y

V

X

V

dY

dP

Z

VW

Y

VV

X

VU

V

∂∂+

∂∂+

∂∂+−=

∂∂+

∂∂+

∂∂+

∂∂ υ

τ (15)

Z-component

)(

)cos(

2

2

2

2

2

2

Z

W

Y

W

X

W

dZ

dPRa

Z

WW

Y

WV

X

WU

W

∂∂+

∂∂+

∂∂+

−Φ=∂∂+

∂∂+

∂∂+

∂∂

υ

τ (16)

Energy equation

)(2

2

2

2

2

2

Z

T

Y

T

X

T

Z

TW

Y

TV

X

TU

T

∂∂+

∂∂+

∂∂=

∂∂+

∂∂+

∂∂+

∂∂

τ (17)

3.4. Vorticity Transport Equation

The governing equations for constant–property,

incompressible flow may be cross differentiated and

subtracted to eliminate pressure term, this introducing the

three components of vorticity. The flowing dimensionless

vorticity equations are obtained as:

( ) ( )Y

TRauV

∂∂Φ+Ω∇=∇Ω−Ω∇+

∂Ω∂

cos..Pr

11

2

11

1r

τ (18)

( ) ( )

X

TRa

Z

TRavV

∂∂Φ−

∂∂Φ+Ω∇=∇Ω−Ω∇+

∂Ω∂

cos

sin..Pr

12

2

22

2r

τ (19)

( ) ( )Y

TRawV

∂∂Φ−Ω∇=∇Ω−Ω∇+

∂Ω∂

sin..Pr

13

2

33

3r

τ (20)



The components of dimensionless vorticity used here are

defined as the curl of the dimensionless velocity in term of

dimensionless coordinates

Vvv

×∇=Ω (21)

Z

V

Y

W

∂∂−

∂∂=Ω1 (21a)

X

W

Z

U

∂∂−

∂∂=Ω 2 (21b)

Y

U

X

V

∂∂−

∂∂=Ω3 (21c)

To calculate the velocity from the vorticity, it is convenient

to introduce a vector potential Ψv

, which may be looked upon

as the three dimensional counterpart of two dimension stream

function. The components of the velocity are related to the

components of dimensionless vector potential. The

dimensionless vector potential is defined such that its curl

equals the dimensionless velocity vector:

Ψ×∇=vv

V (22)

ZYU

∂Ψ∂

−∂Ψ∂

= 23 (23)

XZV

∂Ψ∂

−∂Ψ∂

= 31 (24)

YXW

∂Ψ∂

−∂Ψ∂

= 12 (25)

The vector potential that satisfies the continuity equation is

written as:

)( Ψ×∇⋅∇=⋅∇vv

V (26)

and it must be solenoid, i.e.

02

1

2

2

1

2

2

1

2

=∂

Ψ∂+

∂Ψ∂

+∂

Ψ∂ZYX

(27)

The vorticity is then related to the vector potential as

follows:

2

1

2

2

1

2

2

1

2

1ZYX ∂Ψ∂

+∂

Ψ∂+

∂Ψ∂

=Ω− (28)

2

2

2

2

2

2

2

2

2

2ZYX ∂Ψ∂

+∂

Ψ∂+

∂Ψ∂

=Ω− (29)

2

3

2

2

3

2

2

3

2

3ZYX ∂Ψ∂

+∂

Ψ∂+

∂Ψ∂

=Ω− (30)

3.5. Boundary Conditions

The boundary conditions of the system shown in Figure1 is

detailed as:

Temperature

At 0=Z 10 ≤≤ X and 5.00 ≤≤ Y , 1=T

At 2=Z 10 ≤≤ X and 5.00 ≤≤ Y , 0=T

At 0=X and 1=X 20 ≤≤ Z and 5.00 ≤≤Y , 0=T

At 0=Y and 5.0=Y 20 ≤≤ Z and 10 ≤≤ X , 0=T

Vector potential

At 0=Z 10 ≤≤ X and 5.00 ≤≤ Y

03

21 =∂Ψ∂

=Ψ=ΨZ

At 2=Z 10 ≤≤ X and 5.00 ≤≤ Y

0321 =Ψ=Ψ=Ψ

At 0=X and 1=X 20 ≤≤ Z and 5.00 ≤≤ Y

0321 =Ψ=Ψ=Ψ

At 0=Y and 5.0=Y 20 ≤≤ Z and 10 ≤≤ X

0321 =Ψ=Ψ=Ψ

Vorticity Vector

At 0=Z 10 ≤≤ X and 5.00 ≤≤ Y

Z

V

∂∂−=Ω1

, Z

U

∂∂=Ω 2

, 03 =Ω

At 2=Z 10 ≤≤ X and 5.00 ≤≤ Y

0321 =Ω=Ω=Ω

At 0=X and 1=X 20 ≤≤ Z and 5.00 ≤≤ Y

0321 =Ω=Ω=Ω

At 0=Y and 5.0=Y 20 ≤≤ Z and 10 ≤≤ X

0321 =Ω=Ω=Ω

4. Numerical Solution

The numerical simulation of three-dimensional transient

natural convection flow from inclined isothermal plate is done

using (ADI) method for vorticity and temperature equations,

and by (SOR) method for the vector potential equations.

The energy and vorticity equations (17-20) can be written as

follow:

Z

TW

Z

T

Y

TV

Y

T

X

TU

X

TT

∂∂−

∂∂+

∂∂−

∂∂+

∂∂−

∂∂=

∂∂

2

2

2

2

2

2

τ (31)


1

1

2

1

2

1

2

1

2

1

2

1

2

1

SZ

WZ

pr

YV

Ypr

XU

Xpr

+∂Ω∂

−∂

Ω∂+

∂Ω∂

−∂

Ω∂+

∂Ω∂

−∂

Ω∂=

∂Ω∂τ

(32)

2

2

2

2

2

2

2

2

2

2

2

2

2

2

SZ

WZ

pr

YV

Ypr

XU

Xpr

+∂Ω∂

−∂

Ω∂+

∂Ω∂

−∂

Ω∂+

∂Ω∂

−∂

Ω∂=

∂Ω∂τ

(33)

3

3

2

3

2

3

2

3

2

3

2

3

2

3

SZ

WZ

pr

YV

Ypr

XU

Xpr

+∂Ω∂

−∂

Ω∂+

∂Ω∂

−∂

Ω∂+

∂Ω∂

−∂

Ω∂=

∂Ω∂τ

(34)

where

Y

TRaS

∂∂Φ= cos1

X

TRa

Z

TRaS

∂∂Φ−

∂∂Φ= cossin2

Y

TRaS

∂∂Φ−= sin3

The three equations of vector potential components 1Ψ , 2Ψand 3Ψ can be solved at any time-step using a point iterative

successive over relaxation method [18].

)

(1

)1,,()1,,(

),,(),1,(),,1(),,1(),,(

S

b

kjikji

kjikjikjikjikji

+Ψ+Ψ+

Ψ+Ψ+Ψ+Ψ=Ψ

+−

+−+− (35)

A relaxation factor ΨW was defined as follows,

)( )(

),,(),,(),,(

)1(

),,(

sn

kji

n

kji

n

kji

sn

kji W Ψ−Ψ+Ψ=Ψ Ψ+

(36)

Where the counter (s) refers to the number of successive

point iterations performed at the nth time step, and )1(

),,(

+Ψ sn

kji is

the value of the component Ψ at the nth time step after (s+1)

iterations. The value of )1(

),,(

+Ψ sn

kji are resubstituted in to

equation (35) where is then solved with equation (36) until the

following convergence criterion is satisfied:

ε≤Ψ−Ψ∑ + )( )(

),,(

)1(

),,(

sn

kji

sn

kji (37)

where 3

10−=ε ,the relaxation factor ΨW is 1.7 for trial and

error for the present work.

The local Nusselt number is defined as

0=∂∂−=

ZZ

TNu (38)

Equation (38) is solved using forward finite difference for

four points as follow:

)291811(6

1)1,,()3,,()2,,()1,,( jijijiji TTTT

ZNu −+−

∆−= (39)

Average Nusselt number is defined as

∫∫=A

NudAA

Nu1

(40)

This integration can be carried out using trapezoidal rule

[18].

5. Result and Discussion

In the present study, a finite difference method is applied to

find the numerical solution of three-dimensional transient

natural convection from isothermal square plate subject to the

influence of orientation.

Temperature fields, local and average Nusselt Number

distribution over the heated surface are examined for Grashof

numbers and inclination angles ranged from 103 to 5x10

4 and

from 0o to 180

o, respectively. As far as the ambient fluid

characteristics, Pr= 0.7 was considered.

5.1. Temperature Fields

Figures 2 and 3 show that the isothermal lines of the heated

surface at two plans (z-x) and (z-y) at x=1.0 and Grashof

numbers ranged from 104 to 5x10

4. The temperature fields are

assumed to have a symmetric nature with respect to vertical

plane. It’s clear that the temperature gradient above the heated

surface of square plate increases rapidly in the vicinity of free

edge. The isotherms show formation of a boundary layer heat

transfer along the heated surface. The temperature gradients

are concentrated just above the center of the heated surface,

where a thermal plume is formed.



Figure 2. Isotherms plots of square plate at different inclination angles (Gr=104)

As inclination from horizontal increases, the bouncy

component along the surface arises and causes an increase in

the velocity and also a plume shifted completely to the upper

part of the plate. The density of fluid decreases in the region

between the center and upper edge, while the velocity of the

fluid motion continuous to increase. All that leads to

ascending flow after the center, therefore the boundary layer

thickness continues to increase until the upper edge. However,

when square plate is heated at both horizontal facing upward

and downward orientation it has been found that a plume

rising from the heated surface and there is significant change

on temperature field when the orientation is changed. The

small effect of convection mode is existed at low Grashof

number. Further increase of Grashof number, the isothermal

lines becomes distorted, resulting in an increase in heat

transfer rates. It’s clear from Figure3 that the effect of

convection for Grashof number of 5x104 is obvious by

appearance of plume over the heated surface and the effect of

inclination angle on the shape of isothermal lines over the

plate.


Figure 3. Isotherms plots of square plate at different inclination angles (Gr=5x104)

5.2. Local Nusselt Number

The distribution of local Nusselt number for Grashof

number of 103 to 5x10

4 is shown in Figure 4 at facing upward

and downward orientation. At Gr = 103, it’s found that, the

local Nusselt number has very small values because of small

effect of convection mode of hear transfer exists, since low

Grashof number the mode of heat transfer is conduction only.

At Gr=104 and Gr=5x10

4, the increased contribution of

convection is observed and local Nusselt number values

increase at all.

The higher values of local Nusselt number occur at leading

edge of the square plate and reach to maximum value at

vertical position. Depending on the heated surface orientation,

the heat transfer rate of these two orientations shows a

competitive nature. It is observed that, there is a gradual

increase in the local values of Nusselt numbers compared to

the horizontal position and the horizontal upward and

downward orientations yield the lowest heat transfer

coefficients.



Figure 4. Local Nusselt number in the line of symmetry (Gr=103,104,5x104)

5.3. Average Nusselt Number

Based on the results obtained, correlations have been

proposed to evaluate the Nusselt numbers for both upward and

downward orientation as shown in Figure (5). The form of the

proposed correlation takes into account the variation of the

inclination angle as shown in Figure (6). It’s found that, the

average Nusselt number shows fairly large dependence on the


inclination angles of heated square plate at different values of

Rayleigh number. It is also found that, the Nusselt number

values show strong dependence on inclination angle of the

square plate and it reaches to the maximum value at the

vertical position. The average Nusselt number increases by 20%

at the vertical position then decrease with increase of

inclination of plate at downward orientation.

Figure 5. Nusselt number vs. Rayleigh number correlations at different inclination angles



Figure 6. Nusselt number vs. Rayleigh number correlations

5.4. Comparison with Available Previous Work

The validation of the obtained results is made through

comparison with available numerical and experimental data.

Figure 7 shows a comparison between the present study with

available previous literature [3,8] in case of horizontal

position (Φ=0o) facing upward. It can be seen that, a good

agreement is achieved between the present results and the

available experimental results. Figure 8 shows a convergence

between the present numerical result with literature

experimental data [3] at horizontal position (Φ=180o) facing

downward. In case of vertical position (Φ=90o), it’s found that

there is excellence agreement was obtained with the available

numerical data [14], see Figure 9.

Figure 7. Comparison of average Nusselt number with available theoretical

and experimental data (Φ=0o)


and experimental data ((Φ=180o)


data over vertical square plate

6. Conclusion

There is significant effect on temperature filed and Nusselt

number when the orientation of the heated surface is changed.

It’s also observed that, the Nusselt number shows a strong

function of inclination angle. There is a difference in heat

transfer rates between the upward and downward orientation

where the maximum values are obtained at the vertical

position. Finally, the relation between average Nusselt number

and Rayleigh number are correlated and takes into account the

inclination of the heated surface. Finally, the horizontal

upward and downward orientations yield the lowest heat

transfer rates.

Nomenclature

A Area of the heated surface (m2)

g Gravitational vector (m2/s)

h Heat transfer coefficient (W/m2.K)

H Length of the square plate (m)

k Thermal conductivity (W/m.K)

T Dimensionless temperature= (Θ-Θ∞)/(Θw-Θ∞)

U Dimensionless X-component of velocity =uH/α


V Dimensionless Y-component of velocity =vH/α

W Dimensionless Z-component of velocity =wH/α

X Dimensionless X-coordinate =x/H

Y Dimensionless Y-coordinate =y/H

Z Dimensionless Z-coordinate =z/H

Greek Symbols

α Thermal diffusivity of air (m/ s2)

β Coefficient of thermal expansion (K-1

)

υ Kinematics viscosity of air (m /s2)

τ Dimensionless time = tα/H2

Ψv

Dimensionless vector potential

Ω Dimensionless vorticity vector

Θ Temperature (K)

Φ Angle of inclination

Subscripts

∞ Ambient

w Value of surface

Average value

1 Vector component in X-direction

2 Vector component in Y-direction

3 Vector component in Z-direction

Dimensionless Numbers

Gr Grashof number= gβ(Θw-Θ∞)H3/υ2

Ra Rayleigh number=Gr.Pr= gβ(Θw-Θ∞)H3/υα

Nu Local Nusselt number = hH /k

Pr Prandtl number = υ/α

References

[1] Fu, Wu-Shung, Wei-Hsiang Wang, and Chung-Gang Li, An investigation of natural convection in parallel square plates with a heated top surface by a hybrid boundary condition, International Journal of Thermal Sciences 84 (2014): 48-61.

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[3] Goldstein, R. J., and Kei-Shun Lau, Laminar natural convection from a horizontal plate and the influence of plate-edge extensions, Journal of Fluid Mechanics 129 (1983): 55-75.

[4] Lewandowski, W. M., and P. Kubski, Effect of the use of the balance and gradient methods as a result of experimental investigations of natural convection action with regard to the conception and construction of measuring apparatus, Heat and Mass Transfer 18 (1984): 247-256.

[5] Kitamura, Kenzo, and Fumiyoshi Kimura, Heat transfer and fluid flow of natural convection adjacent to upward-facing horizontal plates, International journal of heat and mass transfer 38 (1995): 3149-3159.

[6] Lewandowski, Witold M., et al., Free convection heat transfer and fluid flow above horizontal rectangular plates, Applied energy 66 (2000): 177-197.

[7] Pretot, S., B. Zeghmati, and G. Le Palec, Theoretical and experimental study of natural convection on a horizontal plate, Applied thermal engineering 20 (2000): 873-891.

[8] Rafah Aziz, Instruction system to study free convection heat transfer from isothermal square flat surface, 2000, M.Sc. Thesis, University of Technology, Baghdad.

[9] Martorell, Ingrid, Joan Herrero, and Francesc X. Grau, Natural convection from narrow horizontal plates at moderate Rayleigh numbers, International Journal of heat and mass transfer 46 (2003): 2389-2402.

[10] Kozanoglu, Bulent, and Jorge Lopez, Thermal boundary layer and the characteristic length on natural convection over a horizontal plate, Heat and mass transfer 43 (2007): 333-339.

[11] Chen, T. S., Hwa-Chong Tien, and Bassem F. Armaly. "Natural convection on horizontal, inclined, and vertical plates with variable surface temperature or heat flux." International journal of heat and mass transfer 29 (1986): 1465-1478.

[12] Lewandowski, Witold M. "Natural convection heat transfer from plates of finite dimensions." International journal of heat and mass transfer 34 (1991): 875-885.

[13] Wei, J. J., B. Yu, and Y. Kawaguchi, Simultaneous natural-convection heat transfer above and below an isothermal horizontal thin plate, Numerical Heat Transfer 44 (2003): 39-58.

[14] Ayad K. Hassan, Prediction of three dimensional natural convection from heated disk and rings at constant temperature, Engineering and Technology 5 (2003):229-248.

[15] Ayad K. Hassan, Prediction of three dimensional natural convection from a horizontal isothermal square plate, Journal of Engineering 13 (2006):742-755.

[16] Corcione Massimo, Heat transfer correlations for free convection from upward-facing horizontal rectangular surfaces, WSEAS Transactions on Heat and Mass Transfer 2 (2007): 48-60.

[17] Torrance K.E, Numerical Methods in Heat Transfer, Handbook of Heat Transfer Fundamentals, McGraw-Hill, 2nd Edition, 1985.

[18] Gerald, Curtis F., Patrick O. Wheatley, and Fengshan Bai. Applied numerical analysis. New York: Addison-Wesley, 1989.

Correlations for Nusselt Number in Free Convection...

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